Maximal clique enumeration (MCE) is a fundamental problem in graph theory. It plays a vital role in many network analysis applications and in computational biology. MCE is an extensively studied problem. Recently, Eppstein et al. proposed a state-of-the-art sequential algorithm that uses degeneracy based ordering of vertices to improve the efficiency. In this paper, we propose a new parallel implementation of the algorithm of Eppstein et al. using a new bit-based data structure. The new data structure not only reduces the working set size significantly but also by enabling the use of bit-parallelism improves the performance of the algorithm. We illustrate the significance of degeneracy ordering in load balancing and experimentally evaluate the impact of scheduling on the performance of the algorithm. We present experimental results on several types of synthetic and real-world graphs with up to 50 million vertices and 100 million edges. We show that our approach outperforms Eppstein et al.'s approach by up to 4 times and also scales up to 29 times when run on a multicore machine with 32 cores.